Mean Center and Standard Distance

Introduction: 
The States of Kansas and Oklahoma are facing a dilemma regarding tornado activity, and whether or not building tornado shelters would be beneficial in certain areas. There are arguments concerning tornado patterns have been repetitive over a series of years, supporting the idea that building shelters in areas of high frequency to be necessary. Others oppose the idea, arguing that there are many areas that have no need for shelters and consider it to be a waste of money. In order to develop a solution to the problem, tornado data has been collected and displayed spatially to determine which citizens have a stronger argument. After providing spatial evidence to either prove or disprove tornado shelter as a need, it becomes easier to identify which group of citizens are more warranted in their complaints and actions can then be taken to solve to the problem. 

Methodology: 
In order to come to a consensus on the issue and thus address the problem, it is necessary to locate tornado locations and determine whether or not a spatial pattern exists. To do this, data was collected regarding tornadoes that have occurred between 1995 and 2006 as well as tornadoes from 2007 to 2012, along with their width in feet. By tracking the location of specific tornadoes that have occurred over a series of 17 years, it becomes possible to identify if there is a pattern occurring in terms of location. If a pattern is identified, then citizens who argue for the building of shelters have support for their argument. 

Not only is it necessary to plot locations and widths of tornadoes throughout Kansas and Oklahoma, it is essential to determine the location of central tendency. To find this location the mean center of the all the tornadoes is calculated. Once this is calculated and plotted it becomes evident where the center of tornado activity is located. Although the mean center is effective in determining spatial patterns, the weighted mean center can be calculated for a more reflective representation of tornado activity based of severity. By calculating and plotting a mean center, and weighting it by the width of each specific tornado the initial mean center shifts. The direction in which the mean center shifts after weighting it indicates the areas which experience wider, more dangerous tornadoes. This becomes significantly helpful in not only identifying a pattern in tornado activity but areas which have experienced more severe tornado activity. 

In addition to plotting the mean centers of activity, standard distance is also helpful in spatially representing patterns. The measurement shows tornadoes which are concentrated around the weighted mean center, and is spatially equivalent to the standard deviation. A standard distance of one standard deviation, weighted by the width, was calculated and displayed. This standard distance indicates 68 percent of the tornado locations which are located around the weighted mean center. This meaning that over half the tornadoes which occurred fall somewhere between the weighted mean and the edge of the weighted standard distance. The smaller the radial distance between the weighted mean and weighted standard distance means the more concentrated tornado activity is in a given area. The larger the radial distance represents a large dispersion of activity throughout.

By developing several spatial representations of the data, conclusions can then be made concerning tornado activity in Kansas and Oklahoma. Displaying the data in correspondence to mean center, weighed mean centers, and weighted standard distances helps indicated if any specific patterns exist in tornado activity. After analyzing each spatial representation, hypotheses can be made in terms of identifying areas with higher probability of tornado activity, thus concluded the necessity of tornado shelters.

Results:

The First few maps created display tornado locations for both 1995 to 2006 data as well as the data from 2007 to 2012 represented by their width. In addition to the spatial dispersion of tornado activity, both the mean center and weighted mean center were plotted in order to portray central tendencies in the data. Figure three displays data from both tornado location data sets along with the mean center and weighted mean center for both.

This map is most helpful in determining shifts in tornado activity in regards to both location overtime and severity. The first observation that can be made from this figure is that mean center for all tornadoes 1995 through 2012 are fairly centralized in the area of study. This indicates that there is a fairly even dispersion of tornado activity throughout the entire area of study. This even dispersion seems to have been consistent over time considering the very small shift in activity to the north. After weighting the mean center on width, the point shifts to the south in 1995 to 2006 data, and to the south east in the 2007 to 2012 data. This indicates there is a slight pattern showing more severe tornadoes occur further south in the area of study. 


Figure 1

Figure 2

Figure 3

The next few maps created once again displayed the tornado for both sets of data in the same manner as the previous maps. With these maps, however, the standard distance was taken into account along with the weighted mean centers. The tornado locations were plotted along with the weighted mean center surrounded by a weighted standard distance, of one standard deviation. Figure six displays tornado locations for both data sets in comparison to each other along with both weighted mean centers, and weighted standard distances. 

This map is helpful in the same sense that figure three was helpful, because all the data is displayed together for easy comparison. The patterns this map will help identify are those of tornado concentration. The weighted mean center still indicates a central tendency located in the center of the study area reflecting fairly even activity throughout the study area. However, the addition of the weighted standard distance provides a more reflective representation of occurring patterns. It is clear that more than half of the study area has experienced 68 percent of the entire amount of tornadoes from 1995 to 2012. The concentration of tornadoes are widely dispersed throughout the study area, meaning a wide variety of areas throughout experience tornadoes. Even though the concentration of tornado activity has appeared to centralize in the more recent set of data, it is still apparent that tornado activity has maintained a wide dispersion pattern affecting many locations throughout the study area. 

Figure 4

Figure 5



Figure 6

Not only are these maps helpful in determining patterns in order to prove whether or not tornado shelters are needed, but when analyzed along with the actual data indicates the specific probable risk each location has in terms of experiencing tornadoes. For instance if patterns continue to hold true for the following years, the probability of a tornado can be calculated to predict the risk of tornado activity. All counties have a probable chance of experiencing 7.61 tornadoes 20 percent of the time. Despite the low chance of encountering at least 7 tornadoes it is possible for it to occur. Furthermore, all counties have a probably risk that they will experience 1.76 tornadoes at least 70 percent of the time. This means there is a significant chance that each county will most likely encounter at least one tornado. Although counties like Alfalfa County, Oklahoma have only experience five tornadoes from 2007 to 2012, in comparison to Russell County, Kansas that experienced 25 tornadoes, both counties have the same amount of probably risk for tornado activity. 

Figure 7

Conclusion: 
Based on the overall results gained from analyzing each spatial representation in correspondence with the raw data and probability several observations can be made. It is apparent that the more recent set of data portraying tornado activity shows there has been a decrease in the number of tornadoes overall. Not only did the 2007 to 2012 data show a decrease in tornadoes, it also showed specific areas with higher frequencies in tornado activity, in comparison to the 1995 to 2006 data which showed a more even dispersion of activity throughout the entire area of study. Even with a decrease in tornado activity and more noticeable affected areas in recent data, the weighted mean center and standard distance show that majority of the study area has similar risk for tornadoes. 

The weighted mean is located primarily in the center of the study area indicating there is a fairly even dispersion of tornadoes throughout both states. The standard distance also represents the same idea that most of the study area maintains a similar risk. This idea is supported because a great majority of the standard distance encloses the study area, and there is no clear evidence that there is a more specific area that has a greater frequency in tornado activity. Not only do the maps indicate a similar risk for tornadoes throughout the entire area of study, but probability represents the same idea. There is a 70 percent chance that there were will be at least one tornado in each county, and a 20 percent chance there could be 7. Each county faces the same probability. When putting probability and spatial data of tornado activity into perspective, and a fairly evenly distributed chance of tornadoes is represented, it becomes difficult to predict which areas are the most at risk. 


Data does not specifically reflects any noticeable patterns in tornado activity throughout Kansas and Oklahoma, making  it difficult to conclude whether or not building tornado shelters would be beneficial in specific areas. There is no strong indication of specific areas which would benefit more from shelters over others. Unless specific clusters of tornados, portrayed by the tornado location points, are used as the only deciding factor to determine where to build shelters, there is no other obvious way to indicate the most beneficial areas. However, the tornado locations alone are not a substantial deciding factor, considering the significant changes in locations over time. Those who argue that some areas have higher frequencies need shelters are inaccurate because high frequency areas tend to change over time.  Thus, there are not identifiable areas where tornado shelters would benefit over other areas.  In order to address the issue, shelters would needed to be built at proportional locations throughout Kansas and Oklahoma to accommodate the same possibility every county faces.